The largest root of random Kac polynomials is heavy tailed

We prove that the largest and smallest root in modulus of random Kac polynomials have a non-universal behavior. They do not converge towards the edge of the support of the limiting distribution of the zeros. This non-universality is surprising as the large deviation principle for the empirical measure is universal. This is in sharp contrast with random matrix theory where the large deviation principle is non-universal but the fluctuations of the largest eigenvalue are universal. We show that the modulus of the largest zero is heavy tailed, with a number of finite moments bounded from above by the behavior at the origin of the distribution of the coefficients. We also prove that the random process of the roots of modulus smaller than one converges towards a limit point process. Finally, in the case of complex Gaussian coefficients, we use the work of Peres and Vir{\'a}g [PV05] to obtain explicit formulas for the limiting objects.


Introduction
Consider a random polynomial of the form: where a 0 , . . . , a n are i.i.d. random variables and z (n) 1 , . . . , z (n) n are the complex zeros of P n . These polynomials are often called Kac polynomials. The zeros of these polynomials are known to concentrate on the unit circle of C as their degree tends to infinity under some moment condition on the coefficients. This universal behavior has been studied by many authors since the work of Sparo and Shur [ŠŠ62] and we refer to the book [BRS86] for more precise information on the history of the topic. The most precise result about this convergence was given by Ibragimov and Zaporozhets in [IZ13] where they prove that for any bounded and continuous function f and any ε > 0 This result means that a proportion going to one of the zeros cluster uniformly on the unit circle. It does not prevent a negligible part of the zeros to be real or to be away from the unit circle. In this situation, it is natural to ask if max k |z (n) k | and min k |z (n) k | converge towards 1 as n goes to infinity. In this note we prove that the behavior of the extremal zeros of P n is not universal and that the random variable max |z (n) k | is usually a heavy tailed random variable. We give an upper bound on the number of finite moments depending on the cumulative distribution function of the coefficients at 0.
To study the zeros of P n for large n, we may want to see P n as the partial sum of a random entire series. If we assume that the random variable |a 0 | is non-deterministic and satisfies E(log(1 + |a 0 |)) < ∞, then the entire series P ∞ (z) = ∞ k=0 a k z k has almost surely a radius of convergence equal to 1 and P ∞ is a random non-constant holomorphic function on the unit disk. Hence P ∞ has a countable set of zeros {z (∞) k } which are counted with multiplicity. As P ∞ is almost surely a non-constant analytic function, its zeros are isolated and have no accumulation point inside the open unit disk. This ensures that there is a finite number of zeros inside any compact set in the disk. Theorem 1.1 (Main result). Assume that the random variable a 0 satisfies E(log(1 + |a 0 |)) < ∞, that |a 0 | is not deterministic and that P(a 0 = 0) = 0. Let n ∈ N ∪ {∞} and n has the same distribution as 1/x (n) 1 .
2. There exists three constants C 1 > 0 , r > 0 and A > 0 depending only on the distribution of |a 0 | such that 3. If there exists k ≥ 0, a > 0 and δ > 0 such that 4. Almost surely, the point process χ n = z  The condition P(a 0 = 0) = 0 ensures that the degree of P n is n. Hence the zero set {z (n) k } is well defined. If we remove this condition, the theorem is still true if we take the convention that for a polynomial of degree k < n, z

The random variable x
In the case of real Gaussian coefficients, Majumdar and Schehr proved a similar result to (3) for the largest real root in [SM08]. They use a Taylor expansion of the first intensity function of the real zeros at 0 to prove that the density of x (n) n decays at infinity like 1/t 2 . Notice that the first three points of Theorem 1.1 are valid for any fixed n. The heavy tail behavior of the largest root is not asymptotic. The points (4) and (5) imply that the heavy tail phenomena do not vanish at infinity, namely that x (∞) does not have more finite moments that x (∞) n . The point (4) is a deterministic statement, and is a direct application of Hurwitz's theorem in complex analysis. It will allow us obtain Corollary 1.2 which gives the limit of the point process χ n for complex Gaussian coefficients.
What does this theorem say for some classical distribution of the coefficients? If the distribution of |a 0 | is absolutely continuous with respect to the Lebesgue measure on R + with density g, then the point (3) can be linked to the density g at zero: 1. if g is continuous at 0 and g(0) > 0 then the largest root in modulus has infinite mean; n has an infinite second moment.
These two situations cover most of the classical examples of random variables. Real Gaussian random variables, exponential random variables, Cauchy random variables and many others are covered by the first part of the remark. Radial complex random variables such as complex Gaussian random variables are covered by the second point. This is a consequence of the polar change of coordinates which adds a factor 2πr to the density. This phenomenon is illustrated in Figure 1: when the coefficients are complex, the density of x (n) 1 vanishes at zero. If the distribution of the a k 's is supported in an annulus bounded away from zero, then all the roots lie in an annulus. This can be seen as a consequence of the Gershgorin circle theorem of localization of the eigenvalues of matrices, applied to the companion matrix of the polynomial P n . In this setting, we also know the empirical measure of P n converges deterministically towards the uniform measure on the unit circle [HN08].
This theorem can be surprising if we compare it to similar results in random matrix theory. There is a strong analogy of results and techniques between random polynomials and random matrices. For Ginibre random matrices or Kac random polynomials, the empirical measures converge towards a deterministic measure and the explicit distribution of the eigenvalues (or zeros) can be computed when the coefficients are Gaussian (real or complex). Large deviation principles for the empirical measures were obtained with the same speed and very similar rate functions ([HP00], [BAZ98] for Ginibre, [ZZ10] and [But16] for Kac polynomials). For random matrices, the large deviation principle for the empirical measure is known not to be universal [BC14] and to depend on the tail of the coefficients of the matrix. For random polynomials, the large deviation principle is universal [BZ17]. For random matrices, the fluctuations of the largest eigenvalue at the edge of the limiting distribution have been studied by many authors since [TW94] and have proven to be universal.
When the a k 's are i.i.d. standard complex Gaussian random variables, the zeros of P n form a Coulomb gas in C n (see [ZZ10]) with density of the form This is similar to the eigenvalues of complex Ginibre random matrices where the density of the eigenvalues on C n is of the form (see [Gin65]) They have in common the same interaction between the particles, but the confining term is different. Why does the largest particle have such a different behavior for polynomials and matrices? If we look at the behavior of the confining term in each variable for random polynomials, we see that it grows at infinity like log(|z|) while the confining term for Ginibre is V (z) = |z| 2 /2. We believe that it is a general fact for Coulomb gases: when the confining term is of order log |z| at infinity, the largest particle has a heavy tail and when the confining term is stronger than logarithm, the largest particle should converge towards the edge of the limiting distribution. The potential energy of the largest particle is approximatively − log |z − w|dµ ∞ (w) + V (z), where µ ∞ is the limiting distribution of the particles. Hence, if V grows faster than logarithm, the cost of having a particle far from the support of µ ∞ grows at infinity. On the other hand, if V and − log |z − w|dµ ∞ (w) are of same order, the cost of having a particle far from the support of µ ∞ is finite. To our knowledge, this phenomenon is not treated in the literature. The heuristic above only relies on "energetic" considerations and has not been proved so far. The same phenomenon should appear for real Gaussian coefficients, as the distribution of the zeros of P n [But16] is also very similar to the distribution of the eigenvalues of the real Ginibre ensemble [Ede97]: both form a mixture of Coulomb gases, each gas having a fixed number of particles on the real line.
If we compare the hypotheses of Theorem 1.1 with the one of the main theorem from [BZ17], we see that both rely on the behavior of the distribution of the coefficients at zero. Within the universality class of random Kac polynomials with Gaussian coefficients, the number of finite moments for the largest root is constant. for exponential coefficients with mean 1 (left) and dµ = 1/2πe −|z| dz on C (right).

Complex Gaussian coefficients.
Theorem 1.1 states that x  As a consequence, the set {|z (n) k |} k≥1 has the same law as the set {U is a sequence of i.i.d. uniform random variables on [0, 1]. 2. The smallest root in modulus of P ∞ , z 1 , has a rotationally invariant distribution and its modulus, x (∞) 1 , has a cumulative distribution function defined on (0, 1) given by: This theorem is no more than the combination of the work of Peres and Virág [PV05] with Theorem 1.1. This allows us to compute the exact limit distribution of the smallest modulus of the zeros of P n . Notice that the cumulative distribution function F x (∞) 1 is of order t 2 around zeros, which implies that 1/x has an infinite variance.

Proofs of the results
We start with a lemma that will be essential in the proof of Theorem 1.1.  1 + a 0 )) < ∞.
Proof of the lemma. For every non negative random variable X we have: Those inequalities come from the relation: E(X) = R + P(X ≥ x)dx. Now we use this inequality to the non negative variable We deduce that ∞ k=1 P( |a k | e εk > 1) < ∞ and so we have, thanks to the Borel-Cantelli lemma lim sup |a k | e εk ≤ 1, which implies sup k∈N |a k | e εk < ∞. The reverse implication relies on a similar reasoning, and will not be used in the proof of the theorem.
As the a k are i.i.d. random variables, the distribution of the random polynomials P n and Q n are the same. If {z Proof of 2. Fix n ∈ N ∪ {∞}. The random variable M := sup k≥2 |a k |/e k is almost surely finite. This is obvious for n < ∞ and this is a consequence of lemma 2.1 for n = ∞. There exists K such that P(M < K) > 0. Let us define C 2 = P(M < K) > 0. The key idea of this proof is to use Rouché's theorem [Lan13,p. 181] to show that P n and P 1 (z) = a 0 + a 1 z have the same number of roots in a neighborhood of the origin. Rouché's theorem is the following: if γ is a closed path holomogous to 0 in some open set U such that γ has an interior and f and g are two analytic functions on U such that for any z ∈ γ |f (z) − g(z)| < |f (z)| then f and g have the same number of zeros in the interior of γ. To bound from below the probability that x (n) 1 is smaller than t, we compare it to the modulus of the root of P 1 .
and 2 |a 0 | |a 1 | ≤ t ≥ P P n has exactly one zero in B 0, 2 |a 0 | |a 1 | and 2 |a 0 | |a 1 | ≤ t ≥ P P n and P 1 have the same number of zeros in B 0, 2|a 0 | |a 1 | and We notice that the triangle inequality implies that Let r be a constant such that C 3 = P(|a 1 | > r) > 0, then we obtain and |a 1 | > r and 2 |a 0 | |a 1 | ≤ t and M < K As t < 1/(A + 1) then A t 2 1−t < t, we obtain that the event For t ≤ e −1 K/r+1 = C 1 , we get, using the independence of the a k 's Proof of 3. Assume that there exists k ≥ 0, a > 0 and δ > 0 such that Let X be a non-negative random variable. Then the Fubini theorem implies that Using this along with the point (1) we get This implies that Proof of 4. Thanks to Lemma 2.1, we obtain that the radius of convergence of the random entire function P ∞ is almost surely 1. This implies that, almost surely, P n converges uniformly towards P ∞ on any closed disk D(0, ρ) with radius ρ < 1. In this setting, the almost sure convergence of the zeros of P n inside D(0, ρ) is exactly Hurwitz's theorem [Con78,p. 152] in complex analysis. Hurwitz's theorem is a consequence of Rouché's theorem, which is a consequence of the argument principle. We give a proof of the convergence of the point processes using directly Rouché's theorem. Let (Ω, F, P) be a probability space on which the a k 's are defined. Let N be a negligible set such that for any ω ∈ Ω \ N , P ∞ is a non-constant entire series with radius of convergence one.
Fix ω ∈ Ω \ N and let z (∞) be a zero of P ∞ , with multiplicity β. As the zeros of P ∞ are isolated, for any ε small enough, P ∞ has no other zero than z (∞) in the closed disk D(z (∞) , ε). Thanks to Rouché's theorem, we know that if then P n and P ∞ have the same number of zeros inside D(z (∞) , ε). The inequality (2) is automatically satisfied for n large enough, as we fixed ε such that P ∞ does not have a zero on the boundary of the disk D(z (∞) , ε).
Here we proved that for any zero of multiplicity β of P ∞ , for any ε > 0 sufficiently small, one can find β zeros of P n at a distance at most ε of z (∞) . This implies that any fixed finite number of zeros of P n converges almost surely towards zeros of P ∞ .
Proof of 5. The fact that x

Comments.
Notice that the proof of points (1), (2) and (3) of Theorem 1.1 for finite n does not use the assumption E(log(1 + |a 0 |)) < ∞. These three points are always valid. This assumption is only needed to make sure that, almost surely, the a k 's do not grow faster than e k . An alternative proof of Theorem 1.1 uses Jensen's formula [Lan13, p. 341] for analytic functions. We chose to use an approach based on Rouché's Theorem as it also implies the convergence of the point process of the small roots.